Yes, I've heard that story before but...............itis documented in some circles as a Jagger/Richards composition.
I stayed in touch with Michael Clarke for over twenty years until he drank himself to death about eight years ago and his thoughts were much like yours. Michael thought maybe trading the publishing rights for whatever Keith had of interest to Gram. Michael of course knew gram from The Byrds, FBB, and his credibility in other bands, like Firefall have some credence to his thoughts.
Anyway, whomever would like to take credit for the tune, it is dedicated to gram and justifiably so.
I will do no bartering with Keith, the results aren't favorable.
peace
me